weak model category



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The concept of weak model categories is a relaxation of that of model categories, even weaker than the concept of semimodel categories, but such that it still allows for a rich theory largely analogous to that of actual model categories:

The weak analogue of the construction of the homotopy category of a model category still exists, as do notions of Quillen adjunction and Quillen equivalence.

Also, for example, an analogue of left or right Bousfield localization of model categories still makes sense for weak model categories; and, as a bonus in contrast to the usual case, it does not require the assumption of left or right properness.


A weak model category is a premodel category that satisfies the following two axioms:

  1. Cylinder axiom: Every cofibration AXA\to X from a cofibrant object to a fibrant object admits a relative strong cylinder object

    X AXI AXX,X\sqcup_A X\to I_A X\to X,

    where the left map is a cofibration and its first component XI AXX\to I_A X is an acyclic cofibration.

  2. Path object axiom: Every fibration AXA\to X from a cofibrant object to a fibrant object admits a relative strong path object

    AP XAA× XA,A\to P_X A\to A\times_X A,

    where the right map is a fibration and its first component P XAAP_X A\to A is an acyclic fibration.


Relation to premodel categories

A premodel category can be upgraded to a weak model category as follows.


If a premodel category admits a weak Quillen cylinder, then it is a weak model category.


A weak Quillen cylinder on a premodel category CC is a pair of left adjoint functors I,D:CCI,D\colon C\to C together with the following commutative square of natural transformations of functors CCC\to C:

id Cid C \longrightarrowlimits i Icr ecrid C \longrightarrowlimits j D,cr\begin{matrix} id_C\sqcup \id_C&\mathop{\longrightarrow}\limits^i&I\cr \downarrow\nabla&&\downarrow e\cr id_C&\mathop{\longrightarrow}\limits_j&D,\cr \end{matrix}

where \nabla is the codiagonal, ii is a cofibration, jj is a trivial cofibration, and the first component of ii is a trivial cofibration.

Here a natural transformation λ:FG\lambda\colon F\to G of functors CCC\to C is a cofibration if for any (trivial) cofibration XYX\to Y the map F(Y) F(X)G(X)G(Y)F(Y)\sqcup_{F(X)}G(X)\to G(Y) is a (trivial) cofibration. Likewise, λ\lambda is a trivial cofibration if for any cofibration XYX\to Y the above map is a trivial cofibration.

Reference: Henry 20, Section 6.

This is essentially a reformulation of Cisinski-Olschok theory?.

Relation to model categories

Model categories can be singled out from weak model categories

by adding the following properties:

  1. Every fibrant object admits a strong path object and every cofibrant object admits a strong cylinder object.

  2. All acyclic cofibrations are trivial cofibrations and all acyclic fibrations are trivial fibrations. (Trivial maps are given as data for a premodel category, whereas acyclic (co)fibrations are defined as (co)fibrations that satisfy a right (left) lifting property with respect to the class of cofibrations with cofibrant source (respectively fibrations with fibrant target.)

  3. The two classes of weak equivalence corresponding to the left and right induced semimodel structures coincide.

Relation to combinatorial model categories

Every combinatorial weak model category can be connected to a combinatorial model category by a zigzag of Quillen equivalences.


Last revised on June 12, 2020 at 08:42:40. See the history of this page for a list of all contributions to it.